{"paper_id":"11dfdc7b-8788-4cab-a05f-8d02fdc08459","body_text":"1 \n \nTitle \nA mathematical description of non-self for biallelic genetic systems in pregnancy, transfusion, and \ntransplantation. \n \nAuthor \nKlaus Rieneck  \n \nCorresponding author:  \nKlaus Rieneck  \nDepartment of Clinical Immunology \nLaboratory of Blood Genetics \nRigshospitalet \nCopenhagen University Hospital \nBlegdamsvej 9, DK-2100, Copenhagen, Denmark \nTel: +45 35 45 35 45 \nEmail: kr@rh.dk \n \nKlaus Rieneck ORCHID 0000-0002-7251-5754 \n \nKeywords: Immunology, assay design, pregnancy, blood transfusion, transplantation, biallelic systems \nAbbreviations: cfDNA: cell-free DNA, GB: gigabase, HLA: Human Leucocyte Antigen, HSCT: Human Stem Cell \nTransplantation; Rh: Rhesus, STR: Short Tandem Repeat \nMSC code 92D99 \nThe author has no relevant financial or non-financial competing interests to disclose. \n \n \n \nAbstract \nA central issue in immunology is the immunological reaction against non-self. The prerequisite for a specific \nimmunological reaction is the exposure to the immune system of a non-self-antigen. \nA simple mathematical description of the fraction of a non-self-allele in biallelic systems at the population \nlevel in transfusion/transplantation medicine and in pregnancy is presented.  \nFurthermore, when designing assays, the mathematical descriptions may help estimate the number of \ngenetic markers necessary to obtain a predetermined probability level in detecting non-self-alleles of a \ngiven frequency. For instance, the described equations can be helpful, in the design of assays where the \nnon-self-allele can be detected by analysis of cfDNA in plasma from pregnant women to estimate fetal \nfraction or cfDNA and to monitor changes in cfDNA in plasma of transplantation patients.  \nBesides the equations that describe all non-self-situations in pregnancy and transfusion/transplantation, a \nnovel way of estimating immunogenicity related to allele frequency is proposed. \n \nIntroduction \nA non-self-situation arises when an individual is exposed to an antigen that this individual does not possess. \nSuch a situation would arise in pregnancy where the mother can be exposed to an antigen that the fetus \nhas inherited from its father or by blood transfusion or organ transplantation from an allogeneic donor. The \nterm non-self is here used to encompass cfDNA with a primary sequence that is not found in the pregnant \nwoman nor recipient of transfusion or organ donation. Simple mathematics was used to describe, in \nbiallelic systems, the fraction of all situations with the presence of a non-self-allele irrespective of allele \n.CC-BY 4.0 International licensemade available under a \n(which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is \nThe copyright holder for this preprintthis version posted April 13, 2024. ; https://doi.org/10.1101/2024.04.10.588831doi: bioRxiv preprint \n\n2 \n \nfrequency. The non-self-scenarios in pregnancy and blood donation/transplantation were addressed and \nthe equations presented define the theoretically maximal fraction of situations where an immune response \nmay arise and define all situations where the non-self-allele can be potentially detected by various assays \nbased on primary DNA sequence. The equations presented can be helpful in assay development in biallelic \nsystems and provide general insight. Most blood group antigens are biallelic as is also much other genetic \nvariation. \nMethods for determining fetal fraction have been published, e.g., (Ni et al., 2019). \n \nMaterials and methods \nOnly biallelic systems with alleles p and q were considered. \nBasic mathematics and theoretical deliberations led to the development of three equations describing all \nnon-self-situations in any biallelic system in pregnancy and transfusion/transplantation and based on the \nequations, simple equations were derived for calculating the number of biallelic markers needed to be \ncombined into an assay to reach a given probability of detecting non-self in pregnancy and \ntransfusion/transplantation including the scenario where both donor and recipient are homozygous albeit \nwith different alleles. \nMicrosoft Excel was used to evaluate the equations in simulations in silico by creating 4000 samples in \nHardy-Weinberg equilibrium but with varying allele frequency p of 0.1, 0.2, 0.3, 0.4, and 0.5 respectively. \nThe 4000 samples with the same allele frequency were duplicated and both identical 4000 samples were \nrandomized separately using the SLUMP function in Excel. A 9-digit number between 0 and 1 was \ngenerated and there were no duplicate numbers. The genotypes were coupled to the random number and \nsorted by the number thus generating a column of randomly sorted genotypes. For the prenatal testing, a \ncolumn with only a single allele was randomized as the paternal contribution to the fetus. The two columns \nof the independently randomized 4000 samples – with the same allele frequency - were combined into one \nset of 4000 outcomes and the number of times, a non-self-situation was created, was counted. This was \nrepeated for a total of 3 times for each allele frequency. The number of non-self-outcomes counted was \ncompared with the number of non-self-outcomes predicted as calculated by equations (3), (9) and (12). \nThe confidence intervals were calculated in GraphPad Prism 10.1.2. \nThe calculations have the Hardy-Weinberg principle as the basis. Frequency-wise in a biallelic system, the \nfrequency of the genotypes in a population is (p+q)2=p2+q2+2pq=1, when there is Hardy-Weinberg \nequilibrium (Hardy, 1908; Weinberg, 1908). \n \n \n \nResults \nIn situations of non-self in pregnancy, the fetus has inherited an allele from its father that the mother does \nnot have. We here extend the non-self term to cover genetic sequence variants that the mother does not \nhave. Irrespective, cfDNA from both the fetus and the mother is present in maternal plasma and non-self \ncfDNA sequences can be used as identification tags for the presence of fetally derived cfDNA in pregnant \nwomen. Such considerations would also apply to transplantation situations albeit the donor has \ncontributed two alleles. \nIn the pregnancy situation, the paternal allele inherited by the fetus may or may not be different from the \nalleles of the mother. In case the fetus inherits a p allele from the father, pf (the f suffix solely denotes that \nthe allele is inherited from the father, it is– indiscernible from other p alleles -) (Fig 1). \n \n.CC-BY 4.0 International licensemade available under a \n(which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is \nThe copyright holder for this preprintthis version posted April 13, 2024. ; https://doi.org/10.1101/2024.04.10.588831doi: bioRxiv preprint \n\n3 \n \np2pf pqpf \npqpf q2pf \nFigure 1 \n \nAnd in case the fetus inherits a q allele (qf) from the father (Fig 2). \n \np2qf pqqf \npqqf q2qf \nFigure 2 \nFigure 1 and 2. The maternal non-self-situations in pregnancy visualized.  \nThe two situations, p2q and q2p are not the same, but both represent a non-self-situation that can provide \nthe necessary. Still, by no means the sufficient basis for immunization and both situations can be \ninformative as to the presence of fetal cfDNA in maternal plasma, i.e., fetal cfDNA that is qualitatively \ndifferent from the maternal allotype as to the primary DNA sequence. \nThis means that for a given non-self-situation both the p2qf and the q2pf outcomes are relevant. \nMathematically, given that p+q=1, the first situation translates to  \nSp=p2q =-p3+ p2       (1) \nThis formula gives the fraction of pregnancies for a given biallelic system where one allele has the \nfrequency p where non-self is present, and the genetic precondition is fulfilled for a possible immunization \nevent (Fig 3). \n \n \n \n \n.CC-BY 4.0 International licensemade available under a \n(which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is \nThe copyright holder for this preprintthis version posted April 13, 2024. ; https://doi.org/10.1101/2024.04.10.588831doi: bioRxiv preprint \n\n4 \n \n0.0 0.5 1.0\n0.0\n0.1\n0.2\n0.3\n0.4\n0.5\n0.6\n0.7\n0.8\n0.9\n1.0\nMaternal allele frequency p\nFraction of non-self situations\n \nFig 3. The fraction of pregnancies with non-self of a single allele in a biallelic system, p2q (-p3+p2) (full line). \nBy integrating equation (1) and calculating the area under the graph in Fig 3, maximally 1/12 of all \npregnancies for all values of p have this genetic constellation that is a precondition for maternal \nimmunization for a given single biallelic antigen system. The maximal pregnancy risk allele frequency is \nfound when p is close to  \n2\n3  and the risk is low both when p is either very low or very high. This means that \nblood group systems where immunization is observed should be most prevalent for blood groups with an \nallele frequency around this maximum, all other things being equal. \nBut p2q only describes half the possible situations. The formula for the converse situation:  \nSq =q2p=(1-p)2p=p3-2p2+ p       (2) \nis depicted with dots in Fig 4. With this distribution, the maximal risk situations are when p is close to \n1\n3. \n.CC-BY 4.0 International licensemade available under a \n(which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is \nThe copyright holder for this preprintthis version posted April 13, 2024. ; https://doi.org/10.1101/2024.04.10.588831doi: bioRxiv preprint \n\n5 \n \n0.0 0.5 1.0\n0.0\n0.1\n0.2\n0.3\n0.4\n0.5\n0.6\n0.7\n0.8\n0.9\n1.0\nMaternal allele frequency p\nFraction of non-self situations\n  \nFig 4. The fraction of non-self in pregnancy in relation to allele frequency for both alleles in a single biallelic \nsystem with the mother being either homozygous p; p2q (full line) or homozygous q; q2p (dotted line). \nThe sum of the two distributions is decisive for the total, theoretical immunization risk in a population for \nthis biallelic locus and for all possible non-self-situations in a population. Determining which distribution, a \ngiven allele belongs to in a biallelic system is not possible.  \nIn the case of an assay for determining either allele, both equation (1) and (2) must be considered. \nTo obtain a high probability for detection of fetal cfDNA in maternal plasma, it is necessary to use several \nbiallelic variant markers to produce an assay to ascertain if one (or more) non-maternal alleles have been \ninherited by the fetus. Such an assay will have the added advantage that no prior knowledge of maternal or \npaternal alleles is needed. An assay addressing this can be useful as a control assay for the presence of fetal \ncfDNA in maternal plasma when making genetic predictions based on findings of fetal cfDNA in maternal \nplasma. \nThe cumulated information from several marker alleles will help establish the presence or non-detectable \npresence, that is, presumed absence or very low level of fetal cfDNA and this information will minimize the \nrisk of a false negative result in relation to other tests based on detection of specific fetal cfDNA. The \nprinciple is illustrated in Fig 5 with an example of  \n \n \n \n \n \n \n \nchr. 1             chr2 chr3            chr4        chr5  \n.CC-BY 4.0 International licensemade available under a \n(which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is \nThe copyright holder for this preprintthis version posted April 13, 2024. ; https://doi.org/10.1101/2024.04.10.588831doi: bioRxiv preprint \n\n6 \n \n \nFig 5. Marker informativity and non-informativity are exemplified. \n \nfive different primer sets targeting five different markers but only the markers on chromosomes 1 and 2 are \ninformative of the presence of fetal cfDNA. So, for one marker either allele may be informative in these \nsituations with both the (p2q) and the (q2p) situations being informative of the presence of fetal cfDNA in \nmaternal plasma. \nIn both situations: the p2qf and the q2pf outcome (1/4 of all outcomes from Punnett squares, grey squares) \nare relevant (Fig 1 and 2) to detect the presence of non-self cfDNA from the fetus. Both situations are \ntheoretically informative (SI), and all other situations are non-informative of the presence of fetal cfDNA in \nmaternal plasma, and without risk of immunization. The equation for the presence of non-self in the \npregnant woman will be: \nSI =(p2q) + (q2p) = -p2+ p        (3) \nThis is shown in Fig 6 for allele frequencies of 0≤p≤1. \n \n0.0 0.5 1.0\n0.0\n0.1\n0.2\n0.3\n0.4\n0.5\n0.6\n0.7\n0.8\n0.9\n1.0\nMaternal allele frequency p\nFraction of non-self situations\n \nFig 6. The fraction of all combined non-self-outcomes for one allele system in pregnancy, shown in relation \nto allele frequency p. \nEquation (3) has a maximum at p=0.5. Thus, alleles with a frequency p=0.5 in a biallelic system are optimal \nfor the detection of the presence of fetal cfDNA, as this allele frequency is most informative. By integrating \nequation (3) and calculating the area under the graph in Fig 6, maximally 1/6 of all situations can be \ninformative in a single biallelic antigen system or result in immunization by contributing a non-self-antigen \nwhen both alleles from a biallelic system are taken into consideration. By using alleles with the most \n.CC-BY 4.0 International licensemade available under a \n(which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is \nThe copyright holder for this preprintthis version posted April 13, 2024. ; https://doi.org/10.1101/2024.04.10.588831doi: bioRxiv preprint \n\n7 \n \ninformative frequencies in the narrow interval between 0.4 to 0.6, potentially 0.0493 can be accessed, that \nis 0.0493/0.1667 ~30% of all theoretically possible information in this setting. \nBy testing 20 alleles with p=0.5, on average about 5 alleles (0.25x20≈5) will be expected to be non-self and \nthus informative for a given blood sample; and at least one allele must be informative for an assay to be of \nuse. \nA non-informative situation, SN can be calculated by \nSN=1-(p-p2) =p2-p+1       (4) \nFor several alleles with differing allele frequencies p1, p2, p3 ...pi an assay, SI(1-i) using these allelic markers \nwill be informative for at least one allelic marker, using equation (4) when. \nSI(1-i)=1-(p12−p1+1)(p22−p2+1)(p32−p3+1)….. (pi2−pi+1)     (5) \nIf all allele frequencies are identical p, then equation (5) can be generalized and the fraction of informative \nsituations testing n alleles can be calculated by \nSI(n)=1-(p2−p+1)n       (6) \nEquation (6) can be rewritten to calculate the number of different allelic markers n, with the same allele \nfrequency p, needed to obtain a desired level of informativity SI(n): \nn =\nln⁡(1−SI(n))\nln⁡(p2−p+1)       (7) \nFor instance, if information on the presence of fetal cfDNA is wanted in 99% of situations of testing a \nmaternal plasma sample using alleles with a frequency of 0.5, it can be calculated that at least 16 biallelic \nmarkers would be needed in an assay as estimated from equation (7).  \nThis is useful information when designing an assay for the detection of fetal cfDNA. \n.CC-BY 4.0 International licensemade available under a \n(which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is \nThe copyright holder for this preprintthis version posted April 13, 2024. ; https://doi.org/10.1101/2024.04.10.588831doi: bioRxiv preprint \n\n8 \n \n0 5 10 15 20 25\n0.0\n0.1\n0.2\n0.3\n0.4\n0.5\n0.6\n0.7\n0.8\n0.9\n1.0\nNumber of different allelic markers\nProbability of detecting a fetal specific allotype\n \nFig 7. The cumulative effect of choosing multiple biallelic markers for detection of the presence of non-self \nin pregnancy i.e., fetal cfDNA in relation to allele frequencies of 0.5 (       ), 0.4 (     ), 0.3 (      ), 0.2 (      ), and \n0.1 (      ), respectively.  \nIn Fig 7 the application of equation (7) shows the effect of the number of markers with different allele \nfrequencies in relation to the probability of detecting a fetal specific allotype. Alleles with allele frequencies \ndown to about 0.3 are highly informative and can be included in an assay. \nIt should be added that if p is substituted with (1-q) in equation (3) -p2+p, as –(1-q)2+(1-q), the result is         \n-q2+q. \n.CC-BY 4.0 International licensemade available under a \n(which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is \nThe copyright holder for this preprintthis version posted April 13, 2024. ; https://doi.org/10.1101/2024.04.10.588831doi: bioRxiv preprint \n\n9 \n \n0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0\n0.0\n0.1\n0.2\n0.3\n0.4\n0.5\n0.6\n0.7\n0.8\n0.9\n1.0\nMaternal allele frequency p\nFraction of all tri-allelic situations\n \nFig 8. All theoretically possible outcomes of allele combinations in pregnancy for a single biallelic marker of \nany frequency: p3 defined by (p3) (lilac graph), q3 defined by (-p3+3p2-3p+1) (green), and the situation with \nnon-self defined by (-p2+p) (red), mother heterozygous and fetus with either allele (-2p2+2p) (black). All \nequations added give 1. (-2p2+2p-p2+p-p3+3p2-3p+1+ p3 = 1) \n \nAn overview of all possible outcomes in pregnancy in relation to allele composition when considering the \ntwo maternal alleles and the allele contributed by the father is shown in Fig 8.  As the mother will invariably \ncontribute one allele to the fetus (except in situations as the recipient of an egg donation, which situation \nwill be the same as described for the transfusion/transplantation situation) only three alleles are \nconsidered. All situations described by equation (3), -p2+p give rise to non-self and consecutively risk of \nimmunization as well as being informative in prenatal assays that detect fetal-specific cfDNA sequences. No \nother situation gives rise to a non-self-situation for the mother. \nIf for instance, the allele frequency of p is 0.5, then 25% of all pregnancies will be at risk of immunization, in \n12,5% of all pregnancies, the mother is homozygous for p and the fetus has received a p from its father, \nand in 12,5% of all pregnancies the mother will be homozygous for q and the father has passed on a q allele \nto the fetus. At p=0.5 then in 50% of all pregnancies, the mother is heterozygous, and in these situations, \nthere is no risk of immunization by the p or q allele. \n \n \n \n \n \n.CC-BY 4.0 International licensemade available under a \n(which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is \nThe copyright holder for this preprintthis version posted April 13, 2024. ; https://doi.org/10.1101/2024.04.10.588831doi: bioRxiv preprint \n\n10 \n \nIn the recipient of transfusion/transplantation, non-self-alleles are introduced in the following 4 situations:  \n1) Recipient*Donor 2) Recipient*Donor 3) Recipient*Donor 4) Recipient*Donor \np2*pq2   p2*q2    q2*pq2    q2*p2 \nThis is shown in Table 1, cells marked with (A) indicating all the non-self-outcomes. \nAll non-self-outcomes can be written as: \ns=p2pq2+p2q2+ q2pq2+ q2p2=p2(p((1-p)2))+p2((1-p)2)+((1-p)2)p((1-p)2)+((1-p)2)p2   (8) \nEquation (8) can be simplified as:  \ns=-2p4+4p3-4p2+2p            (9) \nWhere p is allele frequency. \nAnd for calculation of n markers with an allele frequency of p, cfr. above: \nn =\nln(1−SI(n))\nln(1−2p4−4p3+4p2−2p)      (10) \nSetting SI(n) at 0.99 and p=0.5 gives n≈5.  \nThis equation can be useful for calculating the number of primer sets needed in design for the detection of \nnon-self in this situation where both a heterozygous and homozygous contribution to non-self can be used \nfor assay purposes. \n \n \n \n \n \n \n \nTable 1. Overview of outcomes including non-self-outcomes. A and B are non-self-situations for \ntransfusion/transplant recipients. \n \nIn Table 1. The cells marked (A) define the non-self-situations that must be considered for assay design \nwhen all non-self-situations in transfusion or transplantation must be considered. These situations are \ndescribed as -2p4+4p3-4p2+2p (equation (9)).  \nThe cells marked (B) define the non-self-situations that must be considered when only the two homozygous \nsituations are relevant e.g., in some assays detecting rejection of a transplanted organ. These situations are \ndescribed as 2p4-4p3+2p2 (equation (12)). \nSetting p=0.5, maximally 37.5% of recipients of blood transfusion or a recipient of a donor organ will have a \nnon-self-allele for a given biallelic system, equation (9) and Fig 11. An overview of all the fractions in \ntransfusion and organ donation is shown in Fig 9. \n \n \n \n \ndonor\\recipient p2 q2 pq pq \np2 p2p2 q2p2        (A,B) pqp2 pqp2 \nq2 p2q2       ( A,B) q2q2 pqq2 pqq2 \npq p2pq          (A) q2pq          (A) pqpq pqpq \npq p2pq          (A) q2pq          (A) pqpq pqpq \n \n.CC-BY 4.0 International licensemade available under a \n(which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is \nThe copyright holder for this preprintthis version posted April 13, 2024. ; https://doi.org/10.1101/2024.04.10.588831doi: bioRxiv preprint \n\n11 \n \n10 20 30 40 50 60 70 80 90 100\n0.0\n0.1\n0.2\n0.3\n0.4\n0.5\n0.6\n0.7\n0.8\n0.9\n1.0\nAllele frequency p [%]\nFraction of situations\n \nFig 9. All theoretical situations from transfusion/transplantation in a biallelic system. Situations with non-\nself are defined by (-2p4+4p3-4p2+2p) (red) and are the combined risk of the two homozygous situations \np2q2 and q2p2 and the four situations of non-self for homozygous recipient and heterozygous donor \np2pq+p2pq and q2pq+q2pq (see Table 1). The two latter situations occur with the same fraction as the green \nand orange graphs respectively. The situations without non-self: with a heterozygous recipient and two \ndifferent homozygous donors: pqp2+pqp2 are defined by (-2p4+2p3) (orange) and pqq2+pqq2 are defined by \n(-2p4+6p3-6p2+2p) (green). The situations where the recipient and donor have the same homozygous alleles \nare defined by p4 (p4) (brown), and for q4 by (p4-4p3+6p2-4p+1) (grey). The 4 situations where both recipient \nand donor are heterozygous are defined by (4p4−8p3+4p2) (black). All the above equations added give 1. \n(-2p4+4p3-4p2+2p+ p4+ p4-4p3+6p2-4p+1-2p4+2p3-2p4+6p3-6p2+2p+4p4−8p3+4p2=1) \n \nFrom Fig 9 it seems that non-self-situations (the red graph) arise at a fairly constant level for values of p \nbetween 20% to 80% and make up a fraction of about 0.25-0.35 in this interval of p. None of the other \ngenotype combinations pose any risk as to immunization of the recipient. The two equations for single \nalleles in pregnancy have maximal fractions exactly where the three graphs intersect at p=1/3 and 2/3 \nrespectively. \n.CC-BY 4.0 International licensemade available under a \n(which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is \nThe copyright holder for this preprintthis version posted April 13, 2024. ; https://doi.org/10.1101/2024.04.10.588831doi: bioRxiv preprint \n\n12 \n \n \nIn other situations, where it is desirable to detect admixed DNA from different individuals such as for \nchimerism measurements in HSCT or some cases of organ transplantation it can be desirable to investigate \nonly the double homozygous situation; the mathematics is slightly different. Digital PCR technology may be \nadvantageous in these situations. \nAn optimally informative situation SI for digital PCR to monitor cfDNA in cases of chimerism for instance \nafter transplantation would be when the donor is homozygous for a given marker and the recipient is \nhomozygous for the alternative allele or vice versa (cells marked (B) in table 1): \nSI=p2q2+ q2p2        (11) \nGiven that p+q=1, this can be simplified to \nSI=2p4-4p3+2p2       (12) \nAnd the situation FN that is non-informative \nSN=1-(2p4-4p3+2p2)=1-2p4+4p3-2p2 \nFor more markers with varying allele frequencies p1, p2, p3...pi, where at least one situation is informative. \nSI(1-i) =1-(1- 2p14+4p13-2p12)(1- 2p24+4p23-2p22)(1- 2p34+4p33-2p32).....(1- 2pi4+4pi3-2pi2)   (13) \nWhen p is the same for n different allelic markers, the formula can be simplified to  \nSI(n)=1-(1- 2p4+4p3-2p2)n       (14) \nRearranged:  \nn =\nln⁡(1−SI(n))\nln⁡(1−2p4+4p3−2p2)       (15) \nSetting SI(n) at 0.99 and p=0.5 gives n≈34, i.e., 34 primer sets with biallelic markers with an allele frequency \nof p=0.5 are needed to have, with 99% probability, at least one marker that can be used to discern between \nrecipient and donor cells with a marker that is homozygous in the recipient as well as homozygous in the \ndonor material albeit for the alternative allele. If p=0.4 then 38 allelic markers would be needed to achieve \nthe same SI(n).  \nWith p=0.5 a check of equation (15) for 34 markers gives  \nSI(n)= (1-(2/16))34=0.0107 \n \n \n.CC-BY 4.0 International licensemade available under a \n(which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is \nThe copyright holder for this preprintthis version posted April 13, 2024. ; https://doi.org/10.1101/2024.04.10.588831doi: bioRxiv preprint \n\n13 \n \n0 5 10 15 20 25 30 35 40 45 50\n0.0\n0.1\n0.2\n0.3\n0.4\n0.5\n0.6\n0.7\n0.8\n0.9\n1.0\nNumber of different allelic markers\nProbability of detecting non-self\n \nFig 10. The probability of detecting non-self in relation to allele frequencies and the cumulative number of \nmarkers used in an assay of the double homozygous situation. Allele frequencies of 0.5 (       ), 0.4 (     ),  \n0.3 (      ), 0.2 (      ), and 0.1 (      ), respectively are shown.  \n \nIn Fig 10 the application of equation (14) shows the number of markers with different allele frequencies \nneeded to obtain a given level of probability of detecting non-self. Alleles with allele frequencies down to \nabout 0.4 are highly informative and can be included in an assay.  \n \n.CC-BY 4.0 International licensemade available under a \n(which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is \nThe copyright holder for this preprintthis version posted April 13, 2024. ; https://doi.org/10.1101/2024.04.10.588831doi: bioRxiv preprint \n\n14 \n \n0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0\n0.0\n0.1\n0.2\n0.3\n0.4\n0.5\n0.6\n0.7\n0.8\n0.9\n1.0\nAllele frequency p\nFraction of non-self outcome\n \nFig 11. The fraction of non-self-outcomes predicted by the three equations as a function of allele frequency. \nThe double homozygous non-self-situations defined by (2p4-4p3+2p2) (green), the non-self-situations \nrelevant for pregnancy defined by (-p2+p) (red), and all non-self-situations in transfusion and \ntransplantation are defined by (-2p4+4p3-4p2+2p) (blue). \n \nThe graphs in Fig 11 depict the 3 different scenarios described by the three different equations for the \ncontribution of non-self. At p=0.5, the graph describing the transfusion/transplantation situation has a \nmaximum of 6/16, the graph describing pregnancy has a maximum of 4/16, and the graph describing the \ndouble homozygous situation has a maximum of 2/16, corresponding to the number of squares in table 1 \nwith the genotypes used for the deduction of the equations. \n \n \n \n.CC-BY 4.0 International licensemade available under a \n(which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is \nThe copyright holder for this preprintthis version posted April 13, 2024. ; https://doi.org/10.1101/2024.04.10.588831doi: bioRxiv preprint \n\n15 \n \n10% 20% 30% 40% 50%\n0\n500\n1000\n1500\nAllele frequency p [%]\nNumber of non-self situations\n \n \nFig 12. In silico simulation. Non-self-situations (transplantation, pregnancy, and double homozygous \nscenarios) were counted after in silico simulation of 4000 constructed alleles/genotypes in Hardy-Weinberg \nequilibrium with 5 different allele frequencies (0.1, 0.2, 0.3, 0.4, and 0.5), each with four replicates. The \ncounted non-self-situations (white columns) were compared to the predicted situations (black columns) by \nthe equations (9), (3), and (12). The 95% confidence interval is shown for the counted situations. For each \nallele frequency, the two first columns are from the transfusion/transplantation scenario, the next two \ncolumns are from the pregnancy scenario and the last two columns are from the double homozygote \nscenario. \n \nThe result of the simulation of the combined 2x4000 constructed and randomized genotypes in Hardy-\nWeinberg equilibrium showed good agreement with the results predicted by the equations (Fig 12) and \nTable 2. The number of non-self-situations that were counted from the simulation of the transfusion/organ \nrecipient situation, the pregnancy situation, and the double homozygous scenario, were compared to the \npredicted numbers from the equations (9), (3), and (12). There was no significant difference in Fisher’s \nexact test at p <0.05. \n \n \n \n.CC-BY 4.0 International licensemade available under a \n(which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is \nThe copyright holder for this preprintthis version posted April 13, 2024. ; https://doi.org/10.1101/2024.04.10.588831doi: bioRxiv preprint \n\n16 \n \nAllele \nfrequency found_tx predicted_tx found_pre predicted_pre found_hom predicted_hom \n10% 659 655.2 357 360 64 64.8 \n 644  357  68  \n 669  362  64  \n 637  344  59  \n20% 1068 1075.2 629 640 200 204.8 \n 1089  660  202  \n 1056  628  193  \n 1044  632  200  \n30% 1324 1327.2 840 840 351 352.8 \n 1313  865  328  \n 1324  837  356  \n 1344  824  343  \n40% 1493 1459.2 980 960 458 460.8 \n 1454  985  462  \n 1448  971  442  \n 1444  982  437  \n50% 1512 1500 998 1000 519 500 \n 1510  980  492  \n 1508  996  509  \n 1456  1030  490  \n \nTable 2. The numbers from the simulations and predictions with an allele frequency of p that underlies Fig \n12 are shown. \n \nThe biggest differences were found at 40% allele frequency for the prenatal simulation with a mean of 980 \nand a predicted number of 960. \nThe 95% confidence intervals were calculated based on four generated replicates of random genotype \ncombinations and all except one predicted value (p=40% for pregnancy) were within the 95% confidence \nintervals. \nThe simulation was also done once with 400 samples with consistent results (data not shown). \n \nThe three polynomial equations (9), (3), and (12) can be characterized for all theoretical outcomes valid for \n0≤p≤1 and the fraction ≥0 and ≤1: \n ap4-bp3+cp2 a=2b=c,  0≤a≤16,  1:2:1 \n-ap4+bp3-cp2+dp a=2b=2c=d, 0≤a≤1/0.0625 1:2:2:1      \n       -ap2+bp a=b,  0≤a≤4,  1:1 \nHowever, only equations (9), (3), and (12), are valid in the stated biological context. \n \n \n \n \n.CC-BY 4.0 International licensemade available under a \n(which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is \nThe copyright holder for this preprintthis version posted April 13, 2024. ; https://doi.org/10.1101/2024.04.10.588831doi: bioRxiv preprint \n\n17 \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nFig 13. Proposal of an immunogenicity index related to maximum non-self calculation (not drawn to scale). \n \nFor a quantitative estimation of immunogenicity to enable easy, relative comparability among all biallelic \nserotypes, we propose calculating an immunogenicity index, I=\n𝐵𝐶\nA   (Fig 13). The calculation is simple, and \nthe exposition in the forms of the number of blood transfusions or transplantations or pregnancies should \nbe relatively well documented, it would be more cumbersome to detect and register all immunizations as a \nconsequence of exposition. Each allele can be considered separately.  \nAn index for pregnancy (equation (3) -p2+ p) and transfusion/transplantation (equation (9) -2p4+4p3-\n4p2+2p) should probably be calculated separately. \n \n \nCalculated non-self \nExposition  \nRegistered immunization  \n     A        B                  C \n.CC-BY 4.0 International licensemade available under a \n(which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is \nThe copyright holder for this preprintthis version posted April 13, 2024. ; https://doi.org/10.1101/2024.04.10.588831doi: bioRxiv preprint \n\n18 \n \nDiscussion \nThe premise for the mathematical description of the number of markers needed is that alleles have \nundergone random assortment in accordance with Mendel’s third law which says that alleles are \nindependently assorted and thus that traits encoded by the alleles segregate independently of each other \nduring gamete formation. The physiologic process of meiosis with independent segregation of non-\ndisequilibrium alleles thus underlies the mathematical descriptions. In general, the same assumptions that \napply to Hardy-Weinberg calculations would apply to the mathematical description.  \nAll the non-self-scenarios in biallelic systems were described for pregnancy, transfusion/transplantation, \nand the special situation of a homozygous recipient and a donor homozygous for the alternative allele. In \nthese situations, easy calculation of the number of markers needed for obtaining a desired information \nlevel in assays can be obtained regarding the presence of non-self-genetic variants from equations (7), (10), \nand (15).  \nNon-self-situations are also the prerequisite for an alloimmune response to occur, although importantly \nseveral other factors are needed. \nAssays for determining chimerism in transplantation without prior knowledge of the genotypes of the \ninvolved individuals have been developed (Clausen et al., 2023). For instance, one group has chosen 24 \nindel markers using both homozygous and heterozygous informative marker genotypes (Pettersson et al., \n2021). \nThe equations can be used to assess the number of markers to be used in prenatal control assay for the \npresence of fetal cfDNA to minimize the risk of false negative results. Also, it is important to note that the \ngraph describing the pregnancy situation has a form that indicates that alleles with allele frequencies far \nfrom the optimal 0.5 are very informative as to non-self (Fig 6). This is also indicated in Fig 7 where alleles \nwith a frequency as low as 0.3 appear to be reasonably informative (Lee et al., 2017). \nBy comparing Fig 7 and Fig 10, clearly, more markers are needed in the double homozygous situation to \nobtain the same probability of detecting non-self as compared to the pregnant situation. \nIn the case of the double homozygous situation, a large number of assays with individual markers must be \ndesigned to ensure a useful test with a high rate of useful outcomes. However, the markers should be \nassorted independently and therefore be spaced sufficiently. With a distance between loci of 40 GB (LaRue \net al., 2014) about 75 markers can be designed from the human genome for a single assay. \nDifferent equations describe the fraction of non-self in pregnancy on one hand and transfusion and \ntransplantation on the other hand, which makes biological sense as in pregnancy, the father always passes \nonly one of his two alleles to the fetus. The two scenarios of transfusion and transplantation are analogous \nin respect to the description of non-self. \nIn forensics multiallelic STR systems are routinely used and estimations of the number of biallelic SNPs \nneeded to replace STRs have been done (Amorim and Pereira, 2005; Gill, 2001; Lee et al., 2017). \nWe also suggest a novel way of estimating immunogenicity that may better enable comparison across \ndifferent biallelic systems. The basic idea is to relate immunization to the maximally theoretically possible \nimmunization for any biallelic system and thus enable a different way of comparing immunogenicity among \ndifferent allele systems. Perhaps the calculation of the immunogenicity index may give an alternative \nuniform way of comparing the immunogenicity of different biallelic systems. Once a reliable \nimmunogenicity index has been calculated, it could be used to estimate the completeness of registration of \nimmunization frequency in other populations of similar genetic backgrounds. This could be helpful in the \nregistration of transfusion complications that often involve an antibody response.  Also, in case the \nimmunogenicity index does not vary among different populations, it may give hints as to the biological \norigins of immunogenicity for a given allele. \n.CC-BY 4.0 International licensemade available under a \n(which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is \nThe copyright holder for this preprintthis version posted April 13, 2024. ; https://doi.org/10.1101/2024.04.10.588831doi: bioRxiv preprint \n\n19 \n \nThe suggested immunogenicity index calculation should be evaluated experimentally to gauge the \nrelevance of this approach and the results compared to published data.  \nThe mathematical descriptions were tested in silico to ascertain that the mathematical predictions were \naccurate. There was no significant deviation (at p <0.05) by Fisher’s exact test from the counted versus the \nexpected numbers as calculated by equations (9), (3), and (12), see Fig 12. Thus, this in silico test does not \ninvalidate the predictive accuracy of the equations, however, a more rigorous in silico validation would \nneed both a much larger sample size and many more replicates. In three situations the predicted numbers \nfell just outside the 95% confidence interval, with a total of 4 replicates. With so many calculations and few \nreplicates, this is not surprising. \n \nIn conclusion, a mathematical description is reported of biallelic systems of non-self-allele fractions in 3 \ndifferent scenarios: pregnancy, transfusion/transplantation including the scenario with a homozygous \ndonor and a recipient homozygous for the alternative allele. Also given, are derived equations to calculate \nthe number of marker systems needed to reach a given probability of detecting non-self. These equations \ncan be useful in quantitative estimations for the design of tests for identification purposes e.g., fetal \nfraction or chimerism determination and other purposes. \n \nReferences \nAmorim, A. and Pereira, L., 2005. Pros and cons in the use of SNPs in forensic kinship investigation: a \ncomparative analysis with STRs. Forensic Sci Int. 150, 17-21. \nClausen, F.B., Jorgensen, K.M.C.L., Wardil, L.W., Nielsen, L.K. and Krog, G.R., 2023. Droplet digital PCR-based \ntesting for donor-derived cell-free DNA in transplanted patients as noninvasive marker of allograft \nhealth: Methodological aspects. PLoS One. 18, e0282332. \nGill, P., 2001. An assessment of the utility of single nucleotide polymorphisms (SNPs) for forensic purposes. \nInt J Legal Med. 114, 204-10. \nHardy, G.H., 1908. Mendelian proportions in a mixed population. Science. 28 49–50. \nLaRue, B.L., Lagace, R., Chang, C.W., Holt, A., Hennessy, L., Ge, J., King, J.L., Chakraborty, R. and Budowle, B., \n2014. Characterization of 114 insertion/deletion (INDEL) polymorphisms, and selection for a global \nINDEL panel for human identification. Leg Med (Tokyo). 16, 26-32. \nLee, H.J., Lee, J.W., Jeong, S.J. and Park, M., 2017. How many single nucleotide polymorphisms (SNPs) are \nneeded to replace short tandem repeats (STRs) in forensic applications? Int J Legal Med. 131, 1203-\n1210. \nNi, M., Peng, X.L. and Jiang, P., 2019. Bioinformatics Pipeline for Accurate Quantification of Fetal DNA \nFraction in Maternal Plasma. Methods Mol Biol. 1909, 177-180. \nPettersson, L., Vezzi, F., Vonlanthen, S., Alwegren, K., Hedrum, A. and Hauzenberger, D., 2021. \nDevelopment and performance of a next generation sequencing (NGS) assay for monitoring of \nmixed chimerism. Clin Chim Acta. 512, 40-48. \nWeinberg, W., 1908. Über den Nachweis der Vererbung beim Menschen. Jahresh. Ver. Vaterl. Naturkd. 64 \n369–382. \n \n.CC-BY 4.0 International licensemade available under a \n(which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is \nThe copyright holder for this preprintthis version posted April 13, 2024. ; https://doi.org/10.1101/2024.04.10.588831doi: bioRxiv preprint","source_license":"CC-BY-4.0","license_restricted":false}